Abstract—We consider the problem of increasing the threshold
parameter of a secret-sharing scheme after the setup (share distribution) phase, without further communication between the dealer
and the shareholders. Previous solutions to this problem require
one to start off with a nonstandard scheme designed specifically
for this purpose, or to have communication between shareholders.
In contrast, we show how to increase the threshold parameter of
the standard Shamir secret-sharing scheme without communication between the shareholders. Our technique can thus be applied
to existing Shamir schemes even if they were set up without consideration to future threshold increases.
Our method is a new positive cryptographic application for lattice reduction algorithms, inspired by recent work on lattice-based
list decoding of Reed–Solomon codes with noise bounded in the
Lee norm. We use fundamental results from the theory of lattices
(geometry of numbers) to prove quantitative statements about the
information-theoretic security of our construction. These latticebased security proof techniques may be of independent interest.
Index Terms—Changeable threshold, geometry of numbers, lattice reduction, Shamir secret-sharing.
